Real-time observation of Cooper pair splitting showing strong non-local correlations

Controlled generation and detection of quantum entanglement between spatially separated particles constitute an essential prerequisite both for testing the foundations of quantum mechanics and for realizing future quantum technologies. Splitting of Cooper pairs from a superconductor provides entangled electrons at separate locations. However, experimentally accessing the individual split Cooper pairs constitutes a major unresolved issue as they mix together with electrons from competing processes. Here, we overcome this challenge with the first real-time observation of the splitting of individual Cooper pairs, enabling direct access to the time-resolved statistics of Cooper pair splitting. We determine the correlation statistics arising from two-electron processes and find a pronounced peak that is two orders of magnitude larger than the background. Our experiment thereby allows to unambiguously pinpoint and select split Cooper pairs with 99% fidelity. These results open up an avenue for performing experiments that tap into the spin-entanglement of split Cooper pairs.

meaningless. Thus, I immediately do not understand Fig. 1c, which appears to show correlations dying off in fractions of a msec. Before starting any analysis, the authors need to clarify the data on which they are performing the analysis. Is it the filtered data? (it does not seem to be, at least for the cross correlation) Is it the raw data? (if so, how is the noise taken into account?).
Now to the analysis. The authors use two measures in their analysis, auto-correlation and crosscorrelation, but the convoluted way they do the analysis left me completely confused and with no confidence in their results. First, calculating an autocorrelation of any data set numerically is a fairly straightforward matter. For data with noise, one usually sees a sharp peak near zero which corresponds to the autocorrelation of the noise, which can be taken care of most simply by filtering this high frequency noise in the frequency domain, and then doing an inverse fourier transform. Now the time series data shown in Fig. 2a looks very much like a two-level fluctuator, no doubt because the gates have been tuned to the charge neutrality point. We know for many years now from the Dutta-Horn analysis that the Fourier time of the autocorrelation is a Lorentzian that gives the characteristic fluctuation and the autocorrelation should look like 1 -g2(tau) shown in Fig. 1b, presumably because of the way authors define g2(tau). The characteristic time of the system may be related to the tunneling rates, but it is not clear to me what the complicated analysis based on histograms etc detailed in the supplementary provides that is not given by the simple standard analysis described above.
Analysis of the cross-correlation between the two detectors to pull out the crossed Andreev and elastic co-tunneling events is obviously more difficult, but the analysis in the supplementary again does not inspire confidence. I have already mentioned the issue with the filter time constant above. In spite of this, the authors mention considering correlations after a time scale of 170 usec (again, no explanation or justification given for this time scale). There is a whole discussion of noise and slew rates, all of which could be taken care of simply considering the FT of the raw cross-correlation data as discussed above for the autocorrelation. As to pulling out the CAR and EC events: I would anticipate that an actual measure would involve the cross-correlation as well as the individual autocorrelations..an analysis like this would be of real benefit to the community.
Finally, in the long theoretical analysis in the supplementary, I found no connection to the physics of cross Andreev reflection. It just seems a complicated way to fit the data.
To summarize, while I think the basic result of this work is worth publishing, the entire manuscript needs to be reworked before it is ready for publication. In my opinion, a far more powerful statement can be made by simply showing a series of simultaneous time traces as shown in Fig. 2, with simple statistics on how often various events occur. Even more powerful would be to show that crossed Andreev reflection and elastic cotunneling disappear when the superconductor is no longer superconducting, which can easily be done in the same device by applying a large enough magnetic field.
Reviewer #2 (Remarks to the Author): The paper "Real-time observation of Cooper pair splitting showing strong non-local correlations" by A. Ranni et al. reports on charge correlatioin measurements in a Y-junction device made with a central superconducting eletcrode and two outer metallic contacts. Such devices have attracted a lot of attention in the last years due to their potential for the use of the spin-entangled singlet state as a resource for quantum information. Transport measurements as well as microwave measurements have been carried out in nanowire, nanotube or graphene based systems alike. None of the measurements could explore the real-time correlations. In the spirit of quantum optics experiments, it is crucial to perform time domain correlations. This is what is achieved in the present work, which makes this paper very interesting. I am ready to recommend publication in Nature Communications, but I would like the authors to consider the following comments : 1. The effect of Coulomb interactions is important for the Cooper pair splitting process. However, it has been shown some time ago that interactions can generate positive cross correlations in similar setups but without the superconductor (see A. Cottet et al. PRL 92, 206801 (2004) andPRB 70 115315 (2004)). Although I appreciate that this was mainly for the zero frequency shot noise that this point was made, I would like the authors to discuss how their real time detection scheme enables discrimination from these processes.
2.Have the authors tried to apply a magnetic field to study the normal state ? It was not fully clear to me.
3. In the present work, the timescales probed by the experiment are very slow, so the cohence of Cooper pair splitting cannot be assessed. I think it would ne nice if the authors could put a bit more perspective on the technical evolution which should be done in order to probe the system with the relevant timescales for sensing entanglement and to identify the spin states for the correlated processes.

Response to Reviewers:
We thank both Reviewers for the positive feedback on our manuscript. Reviewer 1 finds that our central findings are "certainly worthy of publication in Nature Communications", and Reviewer 2 writes that the real-time detection of Cooper pair splitting "makes this paper very interesting. I am ready to recommend publication in Nature Communications." In addition, the Reviewers are raising the following issues: Reviewer #1 is concerned about our measurements of cross-correlations on a timescale which is shorter than the detector rise time. However, as we explain below, it is indeed meaningful and possible to perform such measurements. Reviewer #2 is in favor of publishing our manuscript, once we have addressed a couple of issues.
Below, we respond to all concerns, and we attach the revised manuscript.

Response to Reviewer #1:
Reviewer #1: The central finding of this paper, as encapsulated in Fig. 2 of the main text, if it does indeed show what it claims to show, is certainly worthy of publication in Nature Communications. However, there are many questions about the geometry of the device and the execution of the experiment, and it is not clear what the authors hope to prove by their analysis, which in my opinion serve only to obfuscate the results. At the least, the paper requires a complete rewrite.
Our response: We thank the Reviewer for the careful reading of our manuscript and for providing us with several useful comments and suggestions that we respond to below. Let us first reassure the Reviewer that we indeed observe what we claim, namely the splitting of individual Cooper pairs. However, to substantiate this claim, a careful statistical analysis is required. To this end, we show in Figs. 1b and 1c results for the auto and cross-correlation functions, which are similar to what is often measured for photons in quantum optics. This statistical analysis allows us to conclude that two nearly simultaneous tunneling events, as the ones in Fig. 2a, with 99% fidelity correspond to the splitting of a Cooper pair rather than just being two uncorrelated single-electron events. Thus, the statistical analysis in Fig. 1 is an important element in support of our claim. Here, let us also comment on an issue that is raised by the Referee below, namely, if it is possible to measure cross-correlations on a timescale, which is shorter than the rise time of the detectors. This is a critical issue for the results in Fig. 1c, and as we clarify below and in the updated manuscript material, it is indeed possible to measure cross-correlations on such short timescales.
We would also like to point out that the correlation measurements, in particular Fig. 1c, are noteworthy results by themselves, and such measurements are important tools for statistical characterizations. In the following responses, we clarify the concerns about the cross-correlation measurements, and hence we do not see a need to completely rewrite the manuscript. Instead, we have added further details of our analysis, which indeed is sound and valid.
Corresponding changes: We have revised our manuscript so that the electrical circuitry is indicated in Fig. 1a and described in the main text. We have also added a new paragraph and a figure in the Supplemental information (see Fig. 3b) to address the fast time resolution of the cross-correlation function below the detector rise time.
Reviewer #1: Let me first focus on the experimental aspects of the paper. First, a general opinion: since this is primarily an experimental paper, key details of the experiment should be given in the main text, and not buried somewhere in the supplementary materials. For example, simple details of the electronic setup are not given: Is some part of the actual sample grounded? What is the temperature? (there is mention in the Methods that the electronic temperature is 40 mK…is this for all measurements reported in the paper, and how is this determination made?). Also, in this experiment, to borrow a phrase, timing is everything, but key timing details of how the data are taken are given only in the supplementary or not given at all. The statements are sometimes contradictory, and cast doubt on the analysis. I explain in more detail about these issues below.
Our response: We thank the Reviewer for pointing out these issues, and we now provide the additional details about the measurements and the experimental setup. sampling rate of 20 kHz, which corresponds to a point every 50 usec. I presume that the data points on both channels are taken simultaneously using something like simultaneous sampling ADCs, otherwise the whole premise of the paper is moot. The specifications of this instrumentation needs to be given in the main text. There is some mention of "jitter" in the supplementary, but no numbers are given and I don't know what this means. Next, the authors state that the data are digitally filtered with a low-pass filter with a cutoff frequency of 200 Hz. (The authors mention that this restricts the detector rise times to 4 msec, which seems shorter than it should be, but about the right range.) This means that any correlations below a time scale of 5 ms are meaningless. Thus, I immediately do not understand Fig.  1c, which appears to show correlations dying off in fractions of a msec. Before starting any analysis, the authors need to clarify the data on which they are performing the analysis. Is it the filtered data? (it does not seem to be, at least for the cross correlation) Is it the raw data? (if so, how is the noise taken into account?) Our response: In the auto-correlation measurements, the detector rise-time indeed imposes a limitation, since the detector needs to "recover" before the next event can be observed. We refer to this time scale as the dead time. As shown in the Supplementary information, we see this limitation in Fig. 4, where a dip with the width of the detector rise time appears at short times. However, the cross-correlation measurements do not have this limitation. This is because we use two detectors, one for each event and hence the dead time of the detectors is not a limiting factor. Instead, it is rather the noise in the timing of the two detectors that matters. These fluctuations are known as jitter, and we thank the Referee for pointing out that these features are not obvious. We have thus extended the description of the correlation function measurements extensively in to the supplemental material with additional experimental data and a description of how the two detection methods differ. We show now directly based on experimental data how the cross-correlations can be measured on a time scale, which is shorter than the detector rise time. We hope that these additions clarify these issues and makes it clear that our results indeed are sound and correct. For all of our analysis, we have used data, for which high-frequency fluctuations have been filtered out.

Corresponding change:
The timing details including the specifications of the instrumentation are now all provided in the Methods section of the manuscript, including details of the sampling and the sampling rate. We also mention that the time traces are measured simultaneously and that we use the filtered data for all of the analysis. We have also added content to the supplementary material to demonstrate the sub-risetime detection resolution of the crosscorrelations including a new figure with experimental data (see Fig. 3b in the supplementary information) that directly shows how this works. We also briefly describe the fast cross-correlation measurements in the main text.
Reviewer #1: Now to the analysis. The authors use two measures in their analysis, auto-correlation and crosscorrelation, but the convoluted way they do the analysis left me completely confused and with no confidence in their results. First, calculating an autocorrelation of any data set numerically is a fairly straightforward matter. For data with noise, one usually sees a sharp peak near zero which corresponds to the autocorrelation of the noise, which can be taken care of most simply by filtering this high frequency noise in the frequency domain, and then doing an inverse fourier transform. Now the time series data shown in Fig. 2a looks very much like a two-level fluctuator, no doubt because the gates have been tuned to the charge neutrality point. We know for many years now from the Dutta-Horn analysis that the Fourier time of the autocorrelation is a Lorentzian that gives the characteristic fluctuation and the autocorrelation should look like 1 -g2(tau) shown in Fig. 1b, presumably because of the way authors define g2(tau). The characteristic time of the system may be related to the tunneling rates, but it is not clear to me what the complicated analysis based on histograms etc detailed in the supplementary provides that is not given by the simple standard analysis described above.
Our response: We believe that the questions about the cross-correlation measurements above may have caused some concerns about our experimental results. However, let us start by stressing that our analysis of the experimental data in fact is very simple, both for auto-correlation and cross-correlations: Once we detect a tunneling event using one of the detectors, we simply determine, if another tunneling event happens in a short time window around a later time tau. The auto-correlations are obtained by considering tunneling events in the same island, while the crosscorrelations consider tunneling events in different islands. The island at which the second event is considered is the only difference between the two measurements. Otherwise, the analysis is the same in both cases.
Regarding the characteristic time scale of the g2-function in Fig. 1b, the Reviewer is correct that it can be related to the tunneling rates, and that is in fact how we determine the value of 1/4.5 s that enters Eq. (1) of the main text. However, as shown by our theoretical analysis in the supplemental material, the expression for this characteristic time scale is a complicated function of the tunneling rates, see Equations (6,7) of the Supplementary information. The reason for this is that the both islands have several accessible charge states, implying that the system is not just a twolevel fluctuator. Still, as our theoretical analysis shows, the resulting g2-function takes the same form as for a twolevel fluctuator, albeit with a characteristic time scale that is a complicated function of all tunneling rates.
We acknowledge that one may analyze the data from many different angles, including the approach suggested by the Reviewer. However, for our purposes, we are convinced that the standard approach to measurements of g2-functions in quantum optics is the appropriate way to analyze our experiment. Also, as we discuss below, our experimental results are supported by a theoretical model with no free parameters, and the good agreement between theory and experiment makes us further confident in our analysis of the experimental results.
Corresponding change: We have clarified in the supplemental material the key idea how we determine the correlation functions and we stress the similarity between the auto-and cross-correlation measurements as well as explicitly state the only difference between the two. Analysis of the cross-correlation between the two detectors to pull out the crossed Andreev and elastic co-tunneling events is obviously more difficult, but the analysis in the supplementary again does not inspire confidence. I have already mentioned the issue with the filter time constant above. In spite of this, the authors mention considering correlations after a time scale of 170 usec (again, no explanation or justification given for this time scale). There is a whole discussion of noise and slew rates, all of which could be taken care of simply considering the FT of the raw crosscorrelation data as discussed above for the autocorrelation. As to pulling out the CAR and EC events: I would anticipate that an actual measure would involve the cross-correlation as well as the individual autocorrelations..an analysis like this would be of real benefit to the community.
Our response: This comment concerns three different points that we now address in turn: (1) To begin with, we stress that is not more difficult to determine the cross-correlations than the autocorrelations, and they are determined in a similar way as discussed in the response above.
(2) The discussion about the slew rates and noise in the detector is made in the supplemental material to explain the width of the peak in Fig. 1c. The width is given directly by the timing accuracy arising from the detector jitters.
(3) The 170 usec is the bin size that we use for determining the correlation functions. It sets the limit for the time resolution of the distributions and should thus be kept as short as possible. At the same time, the bin size needs to be large enough to collect a reasonable number of counts for each bin, and we have chosen a bin size that balances between these two requirements: It is small enough to resolve the peak structure in Fig. 1c of the main text and large enough for us to collect sufficient statistics with small error bars as illustrated in the figure. Choosing a bin size in this manner is a standard procedure.
However, we thank the Reviewer for bringing up the 170 usec timescale, which led us to think about the choice of bin size more carefully. The sampling rate of 20 kHz gives a time discretization of 50 usec in the detector signals as the Reviewer mentioned earlier. Thus, when looking at time differences between events, the possible results are integer multiples of 50 usec. The ideal way to choose the time window is to make it commensurate with the time separation of the time points. This ensures that each bin has equal number of time points in it and removes a scatter that may arise from having some bins that have one time point more or less than the other bins depending on how the time points fall into the window or out of the time window. We thus changed the time window (i.e. bin size) to 150 usec so that each bin has exactly three time points in it. This improvement led to a correction of the data point at tau = 170 usec in Fig. 1b that was previously too high. This bin had indeed four time points instead of the three that the neighboring data points had. We therefore changed to the commensurate bin size. Otherwise, this improvement made no changes to the resulting figure apart from minor changes one data point.
The authors have made substantial elaborations, particularly in the description of the experimental setup that significantly improve the manuscript. I still believe that the theoretical description and analysis complicates the relatively straightforward result of the experiment, but I am willing to acknowledge that looking at the results from the quantum optics perspective might be valuable.
However, there is still a critical part of the analysis that is faulty, and this is with regard to the minimum time for measuring the cross-correlation functions. As the authors clarify in their response and the updated manuscript, all the analyses are performed on data filtered to give a rise time of 4 ms, including digitization of the signal by a simultaneously sampled data acquisition card. First, I do not understand why the noise of the signal should affect the jitter. The jitter should be determined by the specs of the card, perhaps by the jitter in the clock running the ADCs. It seems the authors are confusing uncertainty in the measurement of the amplitude of the signal with uncertainty in measuring the time, which does not make sense, at least to me.
More important, I do not see how one can get correlation data on time scales less than the filter time of the data. Assume, for simplicity, that a brick wall low pass filter is employed (although I assume that the authors use something more sophisticated). Then the Fourier transform of the crosscorrelation is zero beyond a frequency of 200 Hz, which means that the frequency range over which the cross-correlation is calculated is also terminated at 200 Hz, so that the minimum time over which the discrete real time cross-correlation can be calculated is 4 ms or so. Anything less is not meaningful. I may be wrong, but I would like the authors to show mathematically why this is incorrect.
In the same vein, the authors now also give new information that the data are acquired using preamplifiers with a bandpass of 1 kHz before being digitized. I assume these are low pass filtered: again, I am not sure why the data are digitized at 20 kHz when the data being pre-filtered at 1 kHz? I am also a bit confused by the discussion of the feedback loops used to keep the islands at the charge neutrality point. It seems that these are only used in between measurements to correct for drifts. Are they in operation during the measurements as well? If so, the frequency response of the feedback loops will also come into play in the analysis, unless the time scales are vastly different from the time scales of interest.
Since this is a central point of the paper, I believe it needs to be resolved before the paper is suitable for publication.